PRIMORIAL REFLECTIONS - THEORY #1 - TRIBUTARY GAPS

BY MATT ROTHGERY JAN. 26, 2014

I figured out why we can omit 'even' numbers from a prime number sieve and increment by two's. 

It is because the 1st primorial = 2 and it causes a 'tributary' gap at X minus 2 and X plus 2 for every modulo 2.

But nobody ever generalized this formula because there are only 'odd' and 'even' numbers and '2' forms a nice, discrete pattern! BUT if we apply this formula to the 2nd primorial, 

P_2# = p1  p2 = 2  3 = 6 with tributary gaps X-3 & X+3 for every mod6

generally, P_n# = X demands prime gaps at Xpn for every modX except for the first, thus 'tributary' gaps are a defined class of prime gaps. All other gaps in the prime number distribution must be investigated and defined separately.

Now I propose replacing the terms 'even' and 'odd' with new terms 'biremnal' and 'biremnal+1.' Why?

Our sieves may now and forever - instead of omitting 'even' numbers and incrementing by two's- omit 2, 3, 5, 7-remnal numbers and jump the following pattern:

2,3,5,7([gggQgQgggQgQgggQgggggQgQgggggQ]x6,

gggQgQgggQgQgggggggggQgQgggggg)xINFINITY!

where 'g' indicates a gap and 'Q' is a potential prime number. Of course, higher primorials may be used to superimpose larger gaps but the may be deemed too infrequent to justify implementing into an algorithm for low numbers.

Thanks to James Shelby who suspected such a pattern as I explained my baffling ideas. Whenever I am amazed by my own accomplishments, how can I take the credit? ECCL 2:26